3.104 \(\int \frac{x^{9/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=108 \[ \frac{32 b^3 \sqrt{x}}{5 c^4 \sqrt{b x+c x^2}}+\frac{16 b^2 x^{3/2}}{5 c^3 \sqrt{b x+c x^2}}-\frac{4 b x^{5/2}}{5 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{7/2}}{5 c \sqrt{b x+c x^2}} \]

[Out]

(32*b^3*Sqrt[x])/(5*c^4*Sqrt[b*x + c*x^2]) + (16*b^2*x^(3/2))/(5*c^3*Sqrt[b*x +
c*x^2]) - (4*b*x^(5/2))/(5*c^2*Sqrt[b*x + c*x^2]) + (2*x^(7/2))/(5*c*Sqrt[b*x +
c*x^2])

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Rubi [A]  time = 0.129744, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{32 b^3 \sqrt{x}}{5 c^4 \sqrt{b x+c x^2}}+\frac{16 b^2 x^{3/2}}{5 c^3 \sqrt{b x+c x^2}}-\frac{4 b x^{5/2}}{5 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{7/2}}{5 c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(9/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(32*b^3*Sqrt[x])/(5*c^4*Sqrt[b*x + c*x^2]) + (16*b^2*x^(3/2))/(5*c^3*Sqrt[b*x +
c*x^2]) - (4*b*x^(5/2))/(5*c^2*Sqrt[b*x + c*x^2]) + (2*x^(7/2))/(5*c*Sqrt[b*x +
c*x^2])

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Rubi in Sympy [A]  time = 13.6039, size = 100, normalized size = 0.93 \[ \frac{32 b^{3} \sqrt{x}}{5 c^{4} \sqrt{b x + c x^{2}}} + \frac{16 b^{2} x^{\frac{3}{2}}}{5 c^{3} \sqrt{b x + c x^{2}}} - \frac{4 b x^{\frac{5}{2}}}{5 c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{7}{2}}}{5 c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)/(c*x**2+b*x)**(3/2),x)

[Out]

32*b**3*sqrt(x)/(5*c**4*sqrt(b*x + c*x**2)) + 16*b**2*x**(3/2)/(5*c**3*sqrt(b*x
+ c*x**2)) - 4*b*x**(5/2)/(5*c**2*sqrt(b*x + c*x**2)) + 2*x**(7/2)/(5*c*sqrt(b*x
+ c*x**2))

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Mathematica [A]  time = 0.0327179, size = 52, normalized size = 0.48 \[ \frac{2 \sqrt{x} \left (16 b^3+8 b^2 c x-2 b c^2 x^2+c^3 x^3\right )}{5 c^4 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(9/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[x]*(16*b^3 + 8*b^2*c*x - 2*b*c^2*x^2 + c^3*x^3))/(5*c^4*Sqrt[x*(b + c*x)
])

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Maple [A]  time = 0.006, size = 54, normalized size = 0.5 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ({x}^{3}{c}^{3}-2\,b{x}^{2}{c}^{2}+8\,{b}^{2}xc+16\,{b}^{3} \right ) }{5\,{c}^{4}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2/5*(c*x+b)*(c^3*x^3-2*b*c^2*x^2+8*b^2*c*x+16*b^3)*x^(3/2)/c^4/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 0.740807, size = 197, normalized size = 1.82 \[ \frac{2 \,{\left ({\left (3 \, c^{4} x^{3} - b c^{3} x^{2} + 4 \, b^{2} c^{2} x + 8 \, b^{3} c\right )} x^{3} - 2 \,{\left (b c^{3} x^{3} - 2 \, b^{2} c^{2} x^{2} - 7 \, b^{3} c x - 4 \, b^{4}\right )} x^{2} + 10 \,{\left (b^{2} c^{2} x^{3} + 2 \, b^{3} c x^{2} + b^{4} x\right )} x\right )}}{15 \,{\left (c^{5} x^{3} + b c^{4} x^{2}\right )} \sqrt{c x + b}} + \frac{4 \, b^{3}}{\sqrt{c x + b} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

2/15*((3*c^4*x^3 - b*c^3*x^2 + 4*b^2*c^2*x + 8*b^3*c)*x^3 - 2*(b*c^3*x^3 - 2*b^2
*c^2*x^2 - 7*b^3*c*x - 4*b^4)*x^2 + 10*(b^2*c^2*x^3 + 2*b^3*c*x^2 + b^4*x)*x)/((
c^5*x^3 + b*c^4*x^2)*sqrt(c*x + b)) + 4*b^3/(sqrt(c*x + b)*c^4)

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Fricas [A]  time = 0.228571, size = 69, normalized size = 0.64 \[ \frac{2 \,{\left (c^{3} x^{4} - 2 \, b c^{2} x^{3} + 8 \, b^{2} c x^{2} + 16 \, b^{3} x\right )}}{5 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

2/5*(c^3*x^4 - 2*b*c^2*x^3 + 8*b^2*c*x^2 + 16*b^3*x)/(sqrt(c*x^2 + b*x)*c^4*sqrt
(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(9/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212943, size = 76, normalized size = 0.7 \[ -\frac{32 \, b^{\frac{5}{2}}}{5 \, c^{4}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{c x + b} b^{2} + \frac{5 \, b^{3}}{\sqrt{c x + b}}\right )}}{5 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

-32/5*b^(5/2)/c^4 + 2/5*((c*x + b)^(5/2) - 5*(c*x + b)^(3/2)*b + 15*sqrt(c*x + b
)*b^2 + 5*b^3/sqrt(c*x + b))/c^4